| Abstract: |
| We consider the following nonhomogeneous semilinear fractional Laplacian problem $(-\Delta)^s u+u=\lambda (f(x,u)+h(u))$ in $H^s({\mathbb R}^n)$. We prove that under suitable conditions on $f$ and $h$, there exists $\lambda^{\ast}\in (0,\infty)$ such that the problem has at least two positive solutions if $\lambda\in (0,\lambda^{\ast})$, a unique positive solution if $\lambda=\lambda^{\ast}$, and no solution if $\lambda>\lambda^{\ast}$. We also obtain the bifurcation of positive solutions for the problem at $( \lambda^{\ast},u^{\ast})$ and further analyse the set of positive solutions. |
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